Quantum Markov Order

We formally extend the notion of Markov order to open quantum processes by accounting for the instruments used to probe the system of interest at different times. Our description recovers the classical Markov order property in the appropriate limit: when the stochastic process is classical and the instruments are non-invasive, \emph{i.e.}, restricted to orthogonal, projective measurements. We then prove that there do not exist non-Markovian quantum processes that have finite Markov order with respect to all possible instruments; the same process exhibits distinct memory effects with respect to different probing instruments. This naturally leads to a relaxed definition of quantum Markov order with respect to specified sequences of instruments. The memory effects captured by different choices of instruments vary dramatically, providing a rich landscape for future exploration.